11.5 Day 2 Lines and Planes in Space

Equations of planes

More equations of planes

The angle between two planes• Two distinct planes in 3 dimensional

space are parallel or intersect in a line. If they intersect, you can determine the angle

• (0 < ө < π) by finding the angle between a normal vector for each plane (see the diagram on the next slide)

• Finding the angle between two vectors is done by

Calculating the angle between two planes

Example 3

• Find the general equation of the plane containing the points (2,1,1), (0,4,1) and

(-2,1,4)

Example 4Find the angle between the two planes

and the line of intersection of the two planes

x- 2y + z = 0 and 2x + 3y – 2z = 0

Solution to example 4a

Solution to example 4 bThe line of intersection can be found be found by

simultaneously solving the system of equations

x - 2y + z = 0 multiply the top by -2 and add

2x + 3y - 2z = 0 yields 7y-4z =0 or y = 4z/7

Substitute this into the top equation you can get that x = z/7 set t = z/7 to obtain

x = t, y = 4t, z =7t which is the parametric form of the line of intersection

Example 5

• Find the distance between point Q (1,5,-4)

And the plane given by 3x – y + 2z = 6

Solution to example 5

Example 6

Find the distance between the two parallel planes: 3x – y + 2z - 6 = 0 and

6x – 2y + 4z +4 = 0

Distance between two planes

Solution to example 6

To find the distance between the two planes first choose a point in the first plane say (2,0,0).

Then from the second plane determine that a=6 b=-2 c=4 and d =4 (read these values directly from the second equation)

The distance between the planes is given by:

Example 7

Find the distance between the point

Q (3,-1,4) and the line given by

x = -2 + 3t

y = -2t

z = 1+ 4t

"There are two ways to do great mathematics. The first is to be smarter than everybody else. The second way is to be stupider than everybody else -- but persistent."

-- Raoul Bott

“If I had only one day left to live, I would live it in my statistics class: it would seem so much longer.”